518 



MR. IVORY ON THE EQUILIBRIUM OF A MASS OF 



Stirling in 1735, and by Clairaut in 1737, but only on the supposition of a spheroid 

 little different from a sphere ; and the results obtained by these geometers perfectly 

 coincided with the determination of Newton. In a dissertation on the tides, which 

 shared the prize of the Academy of Sciences of Paris in 1740, Maclaurin made a 

 great addition to the Newtonian theory, by proving that any proposed elliptical 

 spheroid of homogeneous fluid would be in equilibrium if it revolved about its less 

 axis with a certain rotatory velocity, and by introducing in his demonstration accu- 

 rate notions respecting the conditions required for the equilibrium of a fluid entirely 

 at liberty. 



If an oblate elliptical spheroid of homogeneous fluid revolve about the less axis, 

 the equilibrium of the mass will be secured if the resultant of the attractive and 

 centrifugal forces acting upon a particle in the surface be directed perpendicularly 

 towards the surface. In order to prove this, suppose that innumerable surfaces are 

 described within the spheroid, similar to the upper surface, and similarly posited 

 about the centre, and it will be easy to prove with respect to a particle in any of the 

 interior surfaces, that the resultant of its centrifugal force, and of the attraction upon 

 it of all the matter within the surface in which it is placed, is perpendicular to that 

 surface. Now it is proved in the Principia that all the matter between the upper 

 surface and any of the interior surfaces exerts no attraction upon a particle either 

 in or within that surface ; and hence it follows that the resultant of the centrifugal 

 force of a particle, and the attraction upon it of all the matter of the spheroid, is 

 perpendicular to the interior surface passing through the particle. The interior 

 surfaces are therefore the true level surfaces of the spheroid, and the equilibrium of 

 the revolving mass is establishsd by the reasoning in the theorem in No. 5. From 

 this demonstration it would appear that the Newtonian property, according to which 

 the matter of a homogeneous stratum bounded by two similar and concentric ellip- 

 tical surfaces does not attract a particle within the stratum, is not merely accidental 

 to the equilibrium, but a condition necessary to its existence. 



The equilibrium of the oblate spheroid may be made out by a different process. 

 The attraction of the mass upon one of its particles may be investigated ; and, when 

 this done, it is found that the attractions parallel to the equator and perpendicular 

 to the same plane, are proportional to the respective distances of the particle from 

 the axis of rotation and from the equator. It thus appears that the forces urging 

 any particle are known expressions of the coordinates of the point of action ; and 

 therefore the solution of the problem is immediately deduced from the theorem in 

 No. 5. Now in this procedure there is no direct mention made of the Newtonian 

 property; and hence it may, perhaps, be alleged that it is not essential to the 

 equilibrium, although it is a principal step in the former demonstration. But a 

 little reflection will show that the property in question is a condition no less neces- 

 sary in this than in the former investigation ; for it is by means of it that the forces 

 acting upon a particle are disengaged from the upper surface of the fluid, the boundary 



